Phase transitions in one-dimensional nonequilibrium systems

Evans, Martin R.

doi:10.1590/s0103-97332000000100005

Cited by 315 publications

(563 citation statements)

References 63 publications

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“…The particular form of u(n) is motivated by the fact that in the usual ZRP, which corresponds to α = 1, this choice with b > 2 leads to a condensation transition [26]. The rate w with α = 1 represents an interaction between nearest-neighbor sites.…”

Section: Modelmentioning

confidence: 99%

Emergent motion of condensates in mass-transport models

2013

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We examine the effect of spatial correlations on the phenomenon of real-space condensation in driven masstransport systems. We suggest that in a broad class of models with a spatially correlated steady state, the condensate drifts with a nonvanishing velocity. We present a robust mechanism leading to this condensate drift. This is done within the framework of a generalized zero-range process (ZRP) in which, unlike the usual ZRP, the steady state is not a product measure. The validity of the mechanism in other mass-transport models is discussed.

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Section: Modelmentioning

confidence: 99%

Emergent motion of condensates in mass-transport models

2013

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“…Meanwhile, we have from (9) and an application of the divergence theorem that (12) in which the functions ϕ 1 (z) and ϕ 2 (z) relate to the limiting values of ϕ(z) as the point z on the curve C is approached from either side, and the vectorn is the unit vector normal to C at that point. Recall that ϕ(z) is the real part of the complex free energy h(z) and therefore, away from any zeros of the partition function, satisfies the Cauchy-Riemann equations…”

Section: Overview Of Lee-yang Theory Of Equilibrium Phase Transitionsmentioning

confidence: 99%

“…Note that this scenario contrasts with the transfer-matrix approach to one-dimensional equilibrium systems where the partition function is also written as a product of matrices. Since all elements of the transfer matrix are positive the largest eigenvalue cannot become degenerate and therefore there can be no phase transition [12]. However there is no such restriction on the elements of C, and so eigenvalue crossing is permitted and nonequilibrium one-dimensional phase transitions can occur.…”

Section: Iv2 Reaction-diffusion Systems and Directed Percolationmentioning

confidence: 99%

The Lee-Yang theory of equilibrium and nonequilibrium phase transitions

2003

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We present a pedagogical account of the Lee-Yang theory of equilibrium phase transitions and review recent advances in applying this theory to nonequilibrium systems. Through both general considerations and explicit studies of specific models, we show that the Lee-Yang approach can be used to locate and classify phase transitions in nonequilibrium steady states. I IntroductionIn this work we seek a mathematical understanding of phase transitions in the steady state of stochastic many-body systems. Systems at equilibrium with their environment provide examples of such steady states, and the mechanisms underlying equilibrium phase transitions are long known and understood. Experimentally, one can distinguish between two types of transition: the first-order transition at which there is phase coexistence, e.g. between a high-density solid and a low-density fluid, and the continuous transition at which fluctuations and correlations grow to such an extent as to be macroscopically observable.From a thermodynamical perspective, one can understand first-order transitions by associating with each phase a free energy. For a given set of external parameters, the phase 'chosen' by the system is that with the lowest free energy, and so a phase transition occurs when the free energies of two (or more) phases are equal. The sudden changes in macroscopically measurable quantities that take place at first-order transitions are described mathematically as discontinuities in the first derivative of the free energy. Discontinuities in higher derivatives relate to continuous (higherorder) phase transitions.The tools of equilibrium statistical physics allow onein principle at least-to express the free energy solely in terms of microscopic interactions. More specifically, the free energy is given by the logarithm of a partition function, a quantity that normalises the steady-state probability distribution of microscopic configurations. Initially it was not universally accepted that this approach could faithfully describe phase transitions, in particular the first-order solidfluid transition [1]. In order to show that the statistical mechanical approach can reproduce the correct discontinuities in the free energy at a first-order transition, Lee and Yang [2] introduced a description of phase transitions concerning zeros of the partition function when generalised to the complex plane of an intensive thermodynamic quantity. Initially, the theory was couched in terms of zeros in the complex fugacity plane which is appropriate for fluids in the grand canonical (fluctuating particle number) ensemble. However, by mapping a lattice gas onto the Ising model [3], the theory was found to hold equally well for a magnet in a fixed magnetic field. Later [4-6] it became clear that the distribution of zeros in the complex temperature plane can reveal information about phase transitions in the canonical ensemble.In section II we present a brief, self-contained discussion of the Lee-Yang theory of equilibrium phase transitions which relates nonanalytic ...

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“…Equation (2) defines a canonical partition function in that the delta function imposes the constraint of fixed total mass M. As we shall discuss in detail below analysis of (2) has only been carried out in the fluid phase and effectively in the grand canonical ensemble where the total mass is allowed to fluctuate. Our aim here is to present a full analysis of (1,2) in the canonical ensemble. In particular this shall elucidate the mechanism of condensation within factorised steady state.…”

Section: Introductionmentioning

confidence: 99%

“…Although our focus in this paper is on the analysis of the steady state (1,2), it is relevant to briefly review some particular models and dynamics which give rise to such steady states, and previous studies of condensation phenomenon. Firstly we mention the backgammon model [13] where unit masses hop under dynamics respecting detailed balance with respect to an energy function which is simply minus the number of unoccupied sites in the system.…”

Section: Introductionmentioning

confidence: 99%

Canonical Analysis of Condensation in Factorised Steady States

2006

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We study the phenomenon of real space condensation in the steady state of a class of mass transport models where the steady state factorises. The grand canonical ensemble may be used to derive the criterion for the occurrence of a condensation transition but does not shed light on the nature of the condensate. Here, within the canonical ensemble, we analyse the condensation transition and the structure of the condensate, determining the precise shape and the size of the condensate in the condensed phase. We find two distinct condensate regimes: one where the condensate is gaussian distributed and the particle number fluctuations scale normally as L 1/2 where L is the system size, and a second regime where the particle number fluctuations become anomalously large and the condensate peak is non-gaussian. Our results are asymptotically exact and can also be interpreted within the framework of sums of random variables. We further analyse two additional cases: one where the condensation transition is somewhat different from the usual second order phase transition and one where there is no true condensation transition but instead a pseudocondensate appears at superextensive densities.

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Phase transitions in one-dimensional nonequilibrium systems

Cited by 315 publications

References 63 publications

Emergent motion of condensates in mass-transport models

Emergent motion of condensates in mass-transport models

The Lee-Yang theory of equilibrium and nonequilibrium phase transitions

Canonical Analysis of Condensation in Factorised Steady States

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